Reference on continuous-time finite state filtering

Question

Problem:
I'm working in reliability field and have seen papers written on the topic like process of failures when systems are functioning under unobservable (or observable) Markov-like environment, i.e. probability to fail is dependent on the state of environment. This state is described as discrete-state discrete-time homogeneous Markov chain. In mathematical notation it would be like this:
$ P\left [ X_{t}=k | Y_{t}=i \right ]=\pi _{k}\left ( i \right ) $;
where $ X _{t} $ is a binary random failure process with possible states 0 (failed) and 1 (working). And $Y _{t}$ is a Markov process at the moment $t$ being in the $i ^{th}$ state.

My question:

Is it possible (or even reasonable) to extend mentioned model? For example, from discrete-state to continuous-state Markov model? Is there any literature (I havent found yet) about continuous stochastic conditional processes. I suppose it is not so trivial, because for continuouse stochastic process statements like $Y_{t}=i$ are meaningless.

Answer 1

This question is related to the topic of stochastic filtering theory. See e.g. the following monographs * Bucy, Joseph - Filtering for stochastic processes with applications to guidance * Bain, Crisan - Fundamentals of stochastic filtering * Kallianpur - Stochastic filtering theory

Explicit solutions exist for the linear case (Kalman-Bucy filter). For the nonlinear case the situation is more complicated. See also the wikipedia page http://en.wikipedia.org/wiki/Kalman_filter#Kalman.E2.80.93Bucy_filter

Answer 2

What you're looking for is the Wonham Filter. In this setting, $(X_t)_{t\geq 0}$ is a time homogeneous Markov chain defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ taking values in a finite state space $S\triangleq \left\{s_n\right\}_{n=1}^N$ governed by the coupled system: $$ \begin{aligned} p_t^n & \triangleq \mathbb{P}(X_t \in s_n)\\ \frac{dp_t}{dt} & = Q p_t,\\ Y_t & = \int_0^t X_s\beta^{\top}ds + W_t \end{aligned} $$ for some Brownian motion defined on the same probability space as $(X_t)_{t\geq 0}$ and where $\beta\in \mathbb{R}^N$ and $Q$ is an $N\times N$ matrix (typically called the "Q-Matrix") and satisfies: $$ \sum_{i=1}^N q_{i,j}=0, \mbox{ and }q_{i,j}\geq 0 \mbox{ for }i\neq j. $$

You can find the solution to this filtering problem in Theorem 22.3.5 of Sam Cohen's book, as well as many other useful facts like occupation times..in Chapter 22.3.